Formation and Evolution of Planetary Systems in Presence of Highly Inclined Stellar Perturbers Konstantin Batygin1,2, Alessandro Morbidelli1, and Kleomenis Tsiganis3

Astronomy & Astrophysics manuscript no. ms c ESO 2018 October 29, 2018

1 Departement Cassiopee:´ Universite de Nice-Sophia Antipolis, Observatoire de la Coteˆ dAzur, 06304 Nice, France 2 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125 3 Section of Astrophysics, Astronomy & Mechanics, Department of Phyiscs, Aristotle University of Thessaloniki, GR 54 124 Thessaloniki, Greece

Preprint online version: October 29, 2018

ABSTRACT

Context. The presence of highly eccentric extrasolar planets in binary stellar systems suggests that the Kozai effect has played an important role in shaping their dynamical architectures. However, the formation of planets in inclined binary systems poses a considerable theoretical challenge, as orbital excitation due to the Kozai resonance implies destructive, high-velocity collisions among planetesimals. Aims. To resolve the apparent difficulties posed by Kozai resonance, we seek to identify the primary physical processes responsible for inhibiting the action of Kozai cycles in protoplanetary disks. Subsequently, we seek to understand how newly-formed planetary systems transition to their observed, Kozai-dominated dynamical states. Methods. The main focus of this study is on understanding the important mechanisms at play. Thus, we rely primarily on analytical perturbation theory in our calculations. Where the analytical approach fails to suffice, we perform numerical N-body experiments. Results. We find that theoretical difficulties in planet formation arising from the presence of a distant (ã ∼ 1000AU) companion star, posed by the Kozai effect and other secular perturbations, can be overcome by a proper account of gravitational interactions within the protoplanetary disk. In particular, fast apsidal recession induced by disk self-gravity tends to erase the Kozai effect, and ensure that the disk’s unwarped, rigid structure is maintained. Subsequently, once a planetary system has formed, the Kozai effect can continue to be wiped out as a result of apsidal precession, arising from planet-planet interactions. However, if such a system undergoes a dynamical instability, its architecture may change in such a way that the Kozai effect becomes operative. Conclusions. The results presented here suggest that planetary formation in highly inclined binary systems is not stalled by perturbations, arising from the stellar companion. Consequently, planet formation in binary stars is probably no different from that around single stars on a qualitative level. Furthermore, it is likely that systems where the Kozai effect operates, underwent a transient phase of dynamical instability in the past. Key words. Planets and satellites: formation – Planets and satellites: dynamical evolution and stability – Methods: analytical – Methods: numerical

1. Introduction i is the inclination), and libration of its argument of perihelion ω ◦ around ±90 . A necessary criterion for√ the resonance is a suffi- Among the most unexpected discoveries brought forth by a con- ciently large inclination (i > arccos 3/5) relative to the mas- tinually growing collection of extra-solar planets has been the re- sive perturber’s orbital plane, during the part of the cycle where alization that giant planets can have near-parabolic orbits. Since the test-particle’s orbit is circular. the seminal discovery of 16Cygni B (Cochran et al., 1997), fol- lowed by HD80606 (Naef et al., 2001), much effort has been By direct analogy with the Sun-Jupiter-asteroid picture, the dedicated to understanding the dynamical origin and evolution Kozai resonance can give rise to variation in orbital eccentricity of systems with highly eccentric planets. In particular, it has and inclination of an extra-solar planet, whose orbit, at the time been understood that in presence of a companion star on an of formation, is inclined with respect to a stellar companion of inclined orbital plane, the most likely pathway to production the planet’s host star (Wu and Murray, 2003). In the systems arXiv:1106.4051v1 [astro-ph.EP] 20 Jun 2011 of such extreme planet eccentricities is via Kozai resonance mentioned above (16Cygni B, HD80606) the stellar compan- (Eggleton and Kiseleva-Eggleton, 2001). ions’ (e.g. 16Cygni A, HD80607) proper motion has been ver- The Kozai resonance was first discovered in the context of ified to be consistent with a binary solution. Other examples of orbital dynamics of highly-inclined asteroids forced by Jupiter, planets in binary stellar systems are now plentiful (e.g. γ Cephei and has been subsequently recognized as an important process (Hatzes et al., 2003), HD 196885 (Correia et al., 2008), etc) with in sculpting the asteroid belt (Kozai, 1962) as well as being binary separation spanning a wide range (ã ∼ 10 − 1000 AU). the primary mechanism by which long-period comets become However, all planets whose eccentricities are expected to have Sun-grazing (Bailey et al., 1992; Thomas and Morbidelli, 1996). been excited by the Kozai resonance with the companion star Physically, the Kozai resonance corresponds to extensive excur- are in wide binaries. sions in eccentricity and inclination of a test particle forced by a If a Kozai cycle is characterized by a sufficiently small peri- massive perturber,√ subject to conservation of the third Delaunay helion distance, the eccentricity of the planet may subsequently momentum H = 1 − e2 cos(i) (where e is the eccentricity and decay tidally, yielding a pathway to production of hot Jupiters, 2 Batygin et al.: Planet Formation in Inclined Binaries whose orbital angular momentum vector is mis-aligned with re- γ (arcsec/year) spect to the stellar rotation axis (Fabrycky and Tremaine, 2007). The presence of such objects has been confirmed via observa- 2 tions of the Rossiter-McLaughlin effect (McLaughlin, 1924), 0 leading to a notion that Kozai cycles with tidal friction are re- numerical sponsible for generating at least some misaligned systems (Winn -2 et al., 2010; Morton and Johnson, 2011).

Kozai cycles may have also played an important role in - -44 systems where a stellar companion is not currently observed.

Indeed, one can envision an evolutionary history where the bi- perihelion frequnecy -6 nary companion gets stripped away as the birth cluster disperses. In fact, such a scenario may be rather likely, as the majority of - -88 analytical stars are born in binary systems (Duquennoy and Mayor, 1991). -10 a (AU) 16 1820 2224 26 28 30 3232 In this case, a Kozai cycle can be suddenly interrupted, causing semi-major axis (AU) the planet’s eccentricity to become ”frozen-in.” In face of the observationally suggested importance of Kozai Fig. 1. An example of apsidal precession,γ, in a self-gravitating cycles during early epochs of planetary systems’ dynamical evo- disk. Here the disk is assumed to contain 50M⊕ between 16 and lution, the formation of planets in presence of a massive, inclined Sunday,32 May 29, AU, 2011 characteristic of a typical post-formation debris disk in perturber poses a significant theoretical challenge (Larwood et the Nice model of solar system formation (Tsiganis et al., 2005). al., 1996; Marzari et al., 2009; Thebault´ et al., 2010). After all, The solid curve shows the precession rate predicted by eqs. (1)- in the context of the restricted problem (where only the stars (3), as a function of semi major axis. The dots and error bars are treated as massive perturbers), one would expect the proto- show the results of a numerical calculation, integrating 3,000 planetary disk to undergo significant excursions in eccentricity equal-mass particles with a softening parameter of ≈ 0.005 AU and inclination due to the Kozai resonance, with different tem- to smooth the effects of their mutual close encounters. The disk poral phases at different radial distances, resulting in an inco- was binned into 100 annuli in a and the mean frequency of the herent structure. Such a disk would be characterized by high- longitude of pericenter$ ˙ was measured from the time-series of velocity impacts among newly-formed planetesimals, strongly $ of the particles in each bin (dots) as well as its variance (error inhibiting formation of more massive objects (planetary em- bars). Note that the precession frequency of a self-gravitating bryos) (Lissauer, 1993). disk is negative. Damping of eccentricities due to gas-drag has been considered as an orbital stabilization process. However, excitation of mutual inclination among neighboring annuli renders this mech- alytic approach, that gravity is a sufficient mechanism to main- anism ineffective (Marzari et al., 2009). Ultimately, in the contain orbital coherence and planetary growth, without any need text of a restricted model, one is forced to resort to competing to account for other forces acting inside the disk. In fact, we time-scales for formation of planetesimals and dynamical ex- show that one of the primary effects of self-gravity is to induce citation by the companion star. Such an analysis suggests that a fast, rigid recession in the longitudes of perihelion and ascend- although possible, planetary formation in binary systems is an ing node of the disk. This allows for planetary formation to take unlikely event. place, as if the secular perturbations arising from the stellar com- Here, we show that the theoretical difficulties in planet for- panion were not present. It is noteworthy that such a process is mation arising from the companion star, posed by the Kozai ef- in play, for instance, in the Uranian satellite system, where the fect and other secular perturbations, can be resolved by a proper Kozai effect arising from the Sun is wiped out owing to secular account of the self-gravity of the proto-planetary disk (i.e. plan- interactions among the satellites and the precession arising from etesimals embedded in a gaseous disk). During the preparation Uranus’ oblateness (Morbidelli, 2002). of this manuscript, a paper was published (Fragner et al., 2011) Furthermore, we show that even after the formation process addressing the role of the gravity of a gas disk on the relative is complete, and the disk has evaporated, the Kozai effect may motion of embedded planetesimals, with hydrodynamical simu- continue to be wiped out by the orbital precession, arising from lations. The work of Fragner et al. (2011) neglects the gravita- planet-planet interactions. This is again in line with the exam- tional effects of the gas-disk onto itself, and therefore considers ple of the outer solar system, where interactions among the gi- a case where pressure and viscosity keep the disk more coher- ant planets erase a Kozai-like excitation due to the galactic tide ent against external perturbations than would be possible with (Fabrycky and Tremaine, 2007). However, if such a planetary self-gravity alone (Fragner and Nelson, 2010). This parameter system undergoes a dynamical instability, which leads to a con- regime is characteristic of systems where external stresses are siderable change in system architecture, it may evolve to a state strong enough to partially overcome the role of self-gravity, but where the Kozai resonance is no-longer inhibited. not that of the internal forces of the fluid. Examples of such sys- The purpose of this work is to identify the important physical tems include binary stars with moderate separations (60 AU in processes at play, rather than to perform precise numerical sim- the simulations of Fragner et al. (2011)). ulations. Consequently, we take a primarily analytical approach In systems of this sort, the dynamics of the planetesi- in addressing the problem. The plan of this paper is as follows. mals tends to be somewhat different from those of the gas. In section 2, we compute the precession rate, arising from the Consequently, gas-drag induces size-sorted orbital evolutions, self-gravity of the disk and show that it is copiously sufficient to ultimately leading to high-velocity impacts among planetesimals impede the Kozai effect. In section 3, we show that under secu- of different sizes. This again, limits the prospects for accretion. lar perturbations from the companion star, the reference plane of Conversely, in the present paper we consider binary separation the disk precesses rigidly, implying an un-warped structure. In on the order ofa ˜ ∼ 1000 AU, consistent with the cases of 16Cyg section 4, we show how an initially stable two-planet system en- B and HD HD80606. This allows us to show, with a simple an- ters the Kozai resonance after a transient instability causes one Batygin et al.: Planet Formation in Inclined Binaries 3 of the planets to be ejected from the system. We summarize and discuss our results in section 5.

2. Kozai Resonance in a Self-Gravitating Disk We begin by considering the secular dynamics of planetesimals in an isolated, flat nearly-circular disk of total mass Mdisk around a Sun-like (M? = 1M ) star. Due to a nearly-null angular momentum deficit, secular interactions within the disk will not ex- cite the eccentricities and inclinations significantly. Rather, as already mentioned above, the primary effect of disk self-gravity is to induce a fast, retrograde apsidal precession. Our calculation of the induced precession follows the for- malism of Binney and Tremaine (1987), originally developed in the context of galactic dynamics. We work in terms of a po- Fig. 2. Apsidal recession of a self-gravitating disk. The reces- lar coordinate system, where the radial coordinate is logarithmic sion rate, γ is plotted as a function of disk mass. Blue points are ρ r φ ( = ln ) and denotes the polar angle. The reduced potential the model results. The points are well fit by a linear functional due to a disk surface density σ reads: −5 relationship γ = −2.4 × 10 (Mdisk/MJup) rad/year. G Z ∞ Z 2π eρ/2σ − 0 0 Φ = √ p dφ dρ , (1) − × −5 2 −∞ 0 cosh(ρ − ρ0) − cos(φ − φ0) functional relationship γ = 2.4 10 (Mdisk/MJup) rad/year. Having computed the characteristic precession rate, we can now where G is the gravitational constant. We assume σ ∝ r−1. write down the orbit-averaged Hamiltonian of a planetesimal in Consequently, axial symmetry is implicit, and the potential is the disk. only a function of ρ. The characteristic frequencies of a planetes- We work in terms of canonically conjugated action-angle imal in the disk are the mean motion, n, and the radial frequency, Delaunay variables κ: p G = a(1 − e2) , g = ω ! p (4) 1 ∂Φ H = a(1 − e2) cos i , h = Ω n = a ∂r where the inclination i is measured relative to an arbitrary ref- ! ! 3 ∂Φ ∂2Φ erence plane and Ω is the longitude of the node of the disk κ = + (2) a ∂r ∂r2 relative to such a plane. In the analysis that follows, we shall take the binary star’s orbital plane to be the reference plane. The where a is semi-major axis. The apsidal precession that results Hamiltonian is simply from self-gravity, γ, can then be written as the difference be- SG tween the mean motion and radial frequencies K = γG . (5) K SG describes an eccentric precessing orbit on a fixed orbital γ ≡ n − κ. (3) plane (i.e. the plane of the disk). In practice, the calculation of γ is performed by breaking Let us now incorporate the perturbations from the stellar up the disk into cells and computing the derivatives discretely. companion into the Hamiltonian. Due to a considerable orbital Following Levison and Morbidelli (2007), we split the disk into separation between the protoplanetary disk and the perturber, the 1000 logarithmic radial annuli, and take the angular cell width to interactions between the two will be secular in nature. For our be ∆φ = 0.5◦. We assume the disk edges to be a = 0.5AU and purposes here, we take the perturber to lie on a circular, inclined in orbit. A circular perturber implies that, to leading order, potential aout = 50AU, although the results are not particularly sensitive to these choices. The resulting precession in the disk is roughly uni- eccentricity excitations in the disk would arise exclusively from form in a, except for the edges, where this linear theory breaks the Kozai resonance. The free orbital precession induced by the down. Numerical experiments of debris disks, where self-gravity companion can be approximated as (Murray and Dermott, 1999) is taken into account directly, however, show that the precession 3n m˜ a3 g˙ ' (6) rate at the edges is also roughly uniform and quantitatively close free 2 M a˜ to that elsewhere in the disk (see Figure 1). In other words, the m a disk’s apsidal precession is approximately rigid. Consequently, where ˜ and ˜ are the perturber’s mass and semi-major axis and M for the purposes of this work, we take the precession rate evalu- is the mass of the central star. In the following, we assume ated at a = 10AU, to be the characteristic γ for the entire disk. that Generally, typical protoplanetary disks contain Mdisk ∼ 10 − g˙free −|γ| , (7) 100M at the time of formation, in gas and planetesimals. We Jup g have calculated the characteristic precession rate for a planetesi- and neglect ˙free altogether. This condition is satisfied for well- a/a mal embedded in such a disk, for total disk mass range, spanning separated binary systems ( ˜ 1). Note that the lack of orbital eccentricity of the perturber is a mathematical convenience that roughly two orders of magnitude, between Mdisk = 0.1MJup and M = 300M . Figure 2 shows the relationship between γ and makes the calculation more straight forward, without modifying disk Jup our conclusions on Kozai dynamics qualitatively1. Mdisk. Note that unlike typical planetary systems, where secular interactions among planets give rise to positive apsidal preces- 1 An eccentric star would induce some forced eccentricity in the disk. sion, γ of a self-gravitating disk is negative. Quantitatively, for However, because of the fast rigid apsidal precession of the disk, the the assumed disk geometry, the precession rate is well fit by the forced eccentricity will be small and the disk will maintain coherence. 4 Batygin et al.: Planet Formation in Inclined Binaries

45 65 85 imax (deg) 1.0 e sin(g) 1.0 1.0

0.5 0.5 0.5

e cos(g) 0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

0.5 0.5 0.5

1.0 1.0 1.0

1.0 e sin(g) 1.0 1.0

0.5 0.5 0.5

e cos(g) 1 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

0.5 0.5 0.5

1.0 1.0 1.0

1.0 e sin(g) 1.0 1.0

0.5 0.5 0.5

e cos(g) 10 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0

0.5 0.5 0.5 Mdisk (MJup)

1.0 1.0 1.0

Fig. 3. Dynamical phase-space portraits for a planetesimal in protoplanetary disks of various masses, perturbed by a stellar companion at various inclinations showing Kozai resonance. The eccentricity vector is plotted in cartesian coordinates on each panel (x = e cos g, y = e sin g). Regions of libration of argument of perihelion are shown as red curves, while blue curves depict circula- tion. The top panels represent a mass-less disk, middle panels correspond to a Mdisk = 1MJup disk and the bottom panels show a Mdisk = 10MJup disk. Note that the Kozai resonance disappears as the disk mass is increased.

We take the stellar companion’s orbital semi-major axis to ture is implicitly essential to our argument, and we will justify be a = 1000AU and take the inclination as well as disk mass this assumption quantitatively in the next section. to be variable parameters. This choice is motivated by the esti- In accord with the reasoning outlined above, we solely re- mate of the orbital separation between HD80606 and HD80607 tain the Kozai term in the disturbing potential of the stellar com- (Eggenberger et al., 2003). It is noteworthy, that the particular panion. Consequently, the planetesimal’s Hamiltonian now reads choice of a does not have significant consequences on the dy- (Kinoshita and Nakai, 1999) namics of the disk, beyond setting the time-scale on which the Kozai effect operates, provided that it is large enough that con- ( dition (7) is satisfied. In this section, we shall assume that the am˜ n˜2 15(a − G2)(G2 − H2) cos(2g) K SGK = γG + nodal reference plane of the disk precesses rigidly, and the disk (M + m˜ ) G2 remains un-warped. In other words, we assume that no mutual ) (5a − 3G2)(G2 − 3H2) inclination is excited between neighboring disk annuli. This fea- − (8) G2 Batygin et al.: Planet Formation in Inclined Binaries 5 wheren ˜ is the stellar perturber’s mean motion. Notice that in the tial. On the other hand, if the Kozai resonance forces the plan- Hamiltonian above, γ could be factorized, so that the magnitude etesimals to acquire a considerable eccentricity during their evo- of the perturbation would be proportional ton ˜/γ. Since γ is a lin- lution (the cases with low disk mass in Figure 3 or, equivalently, ear function of the disk mass, this illustrates that it is equivalent cases with a close stellar companion) the gas-disk may remain to have a closer binary companion (largern ˜) to a proportionally more circular than the planetesimals, thanks to its additional dis- less massive disk. sipative forces. This is the situation illustrated in Fragner et al. With this simplified dynamical model in place, we can now (2011), where a differential evolution of planetesimals and gas, explore the effect of self-gravity on the orbital excitation of the leads to size-dependent gas-drag forces. planetesimals in the disk due to the Kozai resonance. We study In conclusion, recalling that 10MJup is a lower-bound for the three choices of perturber inclination: i = 45◦, i = 65◦ and i = mass of a typical protoplanetary disk, this analysis suggests that 85◦. To obtain a dynamical portrait of the system, we proceed as planetary formation can take place in well separated binary sys- follows. Because the variable h does not appear in (8), the action tems like 16Cyg and HD80606-7 as if secular perturbations, aris- H is a constant of motion. On each H = constant surface, the ing from the companion star were not present. Hamiltonian, K SGK, describes a one-degree of freedom system, in the variables G, g. Simultaneously, because the Hamiltonian is also a constant of motion, the dynamics is described by the level curves of the Hamiltonian. For simplicity, we show the dynamics 3. Rigid Precession of a Self-Gravitating Disk in cartesian coordinates (x = e cos g, y = e sin g) in Figure 3, where e is computed from the definition of G, for the assumed In the previous section, we showed that a self-gravitating disk is value of a (here a = 10AU). Given that H is constant on each not succeptible to excitation by the Kozai resonance. However, panel, a given eccentricity also yields the inclination. The panels in order for our argument to be complete, it remains to be shown that the assumptions of rigid precession of the disk’s nodal ref- are identified by the value of the inclination imax that corresponds to e = 0 for the given value of H (i.e. the inclination of the star erence plane, as well as the lack of the excitation of mutual in- relative to the initial, circular disk). Similarly, the maximal value clination within the disk, hold true. To justify our assumptions, of e on each panel corresponds to i = 0. it is sufficient to consider a nearly circular self-gravitating disk It is useful to begin with a discussion of a mass-less disk and show that it is characterized by rigid nodal precession, since as this configuration is often assumed in formation studies. The we have already shown that a flat disk will remain circular under corresponding plots are shown as the top panel of Figure 3. In external perturbations. this case, there is no added precession (γ = 0) so the Kozai Intuitively, one can expect a rigidly precessing flat disk, from resonance is present for all considered choices of inclination. adiabatic invariance. Consider an isolated self-gravitating disk The phase-space portraits show that any orbit which starts out at where mutual inclinations (inclinations with respect to the in- low eccentricity (near the origin) will follow a trajectory which stantaneous mid-plane), î, are initially small (i.e. sin î ∼ î 1). will eventually lead to a highly eccentric orbit, regardless of ini- Forced by self-gravity, the mutual inclinations within the disk ◦ tial phase. In particular, for imax = 45 , a particle which starts will be modulated on a characteristic (secular) timescale related ◦ ˆ out on a circular orbit will attain emax ' 0.4. For imax = 65 , to the precession of the longitude of the node, Ω, relative to the ◦ emax ' 0.85 and for imax = 85 , emax ' 1. The resulting high- disk mid-plane. One can define the action, velocity collisions render formation of planetary embryos ineffective. Consequently, one should not expect planets to form un- I der the mass-less disk approximation. J = îdΩˆ = const. (9) Let us now consider a Mdisk = 1MJup disk. The phase-space portraits of this system are shown as the middle panels of Figure 3. Although the quoted value corresponds to a very low-mass which represents the phase-space area bounded by a secular cy- disk, the situation is considerably different from the mass-less cle (Neishtadt, 1984). If the disk is subjected to an external ◦ 2 perturbation (such as the torquing from a stellar companion), case. For imax = 45 , the Kozai resonance is no longer effective . Thus, an initially nearly-circular orbit will retain its near-zero whose characteristic timescale is much longer than the secular eccentricity, allowing for planetary formation to take place. The timescale on which self-gravity modulates the mutual inclina- ◦ ◦ tions (the so-called adiabatic condition), J will remain a con- Kozai resonance still operates in the imax = 65 and imax = 85 served quantity (Henrard and Morbidelli, 1993). This implies cases, but the maximum eccentricities are now lower (emax ' that the mutual inclinations within the disk will remain small. 0.55 and emax ' 0.95 respectively) compared to the mass-less disk scenario. This is true for each annulus of the disk in which the adiabatic Finally, the bottom panels of Figure 2 show the phase-space condition is fulfilled. Consequently, the disk will remain un- portraits of a M = 10M disk. Here, the Kozai resonance warped and the disk mid-plane will precess rigidly at constant disk Jup inclination relative to the binary star’s orbital plane. is completely wiped out, for all values of imax. Particularly, circular orbits remain circular (the center of each panel is a stable The same idea can be illustrated more quantitatively in the equilibrium point). Strictly speaking, these calculations describe context of classical Laplace-Lagrange secular theory. One pos- the dynamics of planetesimals embedded in the disk. However, if sible way to view the inclination dynamics of disk is by model- planetesimals remain circular, the gaseous component of the disk ing the disk as a series of massive self-gravitating rings, adjacent must do so as well because it feels the same gravitational poten- to one-another. Note that the same technique cannot be directly applied to eccentricities, since it would predict a positive apsi- 2 Interestingly, the disappearance of the Kozai separatrix is not ex- dal precession, whereas in reality the apsidal precession would actly symmetric with respect to the sign of γ. If γ is negative, it imme- be negative, as shown in the previous section. This is because, diately acts to erase the Kozai effect. However, a small positive γ (i.e. as soon as the eccentricity is non zero, a ring would start to in- −5 ◦ γ = 10 for imax = 45 ) can act to enhance to Kozai effect. The effect tersect the adjacent rings, violating the assumption on which the however rapidly turns over for faster positive precession (i.e. γ > 10−4). Laplace-Lagrange secular theory is based. 6 Batygin et al.: Planet Formation in Inclined Binaries

ˆi (deg) 180 Mdisk = 10 MJup Mdisk = 0 150

t = 3 Myr 120

100 t = 2 Myr

Myr Myr 3 2 Myr = = 1 t t t = 60 t = 1 Myr 50

0 a (AU) 0 10 20 30 40 50 Fig. 5. Orbital evolution of a two-planet system and its tran- Fig. 4. Inclination structure of a mass-less (blue) and a self- sition into the Kozai resonance via an instability. The figure gravitating Mdisk = 10MJup disks. Here the inclination is mea- shows the semi-major axes, as well as perihelion and aphelion sured relative to the original plane of the disk. The inclination is distances as functions of time. The planets initially start out in shown as a function of semi major axis a at t = 1, 3, and 5 Myr. a metastable configuration which is protected from Kozai reso- Note that the mass-less disk is considerably warped due to the nance by apsidal precession, arising from planet-planet interac- perturbations from the companion star, while the self-gravitating tions. Following ∼ 12Myr of dynamical evolution, the planets disk maintains a uniform inclination. in this case, the growth of suffer a dynamical instability, during which the initially outer inclination with time is due to the rigid precession of the disk planet is ejected. Consequently, the remaining planet enters the relative to the binary star plane. The inclination returns back to Kozai resonance. zero after a precession period.

initial mutual inclination between the disk and the stellar com- The scaled Hamiltonian of a given annulus j, where exclu- panion to be i = 65◦. Indeed, the evaluation of the solution for sively secular terms up to second order in inclination have been various disk masses shows that the disk precesses rigidly, if the retained, reads (Murray and Dermott, 1999) disk mass exceeds Mdisk & 1MJup. This threshold is in rough quantitative agreement with the numerical models of Fragner et N 1 0 X 0 0 0 0 al. (2011). Figure 4 shows the evolution of the inclination as a K LL = B i 2 + B i i cos(Ω − Ω ) (10) j 2 j j j jk j k j k function of semi-major axis, of a massless (blue) disk as well as j=1, j,k a self-gravitating (black) MJup = 10MJup disk at various epochs. where the primed quantities are expressed with respect to an The reference plane for the measure of the inclination is the ini- fixed inertial plane (here the initial plane of the disk). In this tial plane of the disk. We see that for the mass-less disk the in- approach, the disk is broken up into N − 1 annuli whereas the clination varies considerably with semi-major axis, which means Nth index corresponds to the stellar companion. The coefficients that the disk is significantly warped, as one would expect in the B and B take the form context of a standard restricted 3-body problem. However, the j j jk inclination of a self-gravitating disk is nearly constant in semi- N major axis, depicting an unwarped, rigid structure. Note, how- n X m j k (1) ever, that the inclination changes with time. this is because the B j j = − α jkα¯ jkb3/2(α jk) 4 M? + m j k=1,k, j disk is precessing with a constant inclination relative to the plane of the binary star, so that the disk’s inclination relative to its ini- n j mk (1) B jk = α jkα¯ jkb3/2(α jk) (11) tial plane has to change periodically, over a precession period, 4 M? + m j ◦ ◦ from imin = 0 to a maximum of imax = 130 and back. where m j denotes the mass of a given annulus j if j < N, In conclusion, the assumption of untwisted structure, that we mN = m˜ , α = a j/ak,α ¯ = α if perturbation is external and employed in the previous section when calculating the excitation α¯ = 1 otherwise, while b(1) is the Laplace coefficient of the by the Kozai resonance, is valid for massive disks perturbed by 3/2 distant stellar companions, such as the ones that we consider in first kind. Similarly to equation (6), the diagonal terms in the B this paper. This conclusion does not apply only to the planetesi- matrix correspond to the free nodal precession rates. To crudely mal disk, but also to the gas disk, for the same reasons mentioned account for the large mutual inclination of the stellar compan- at the end of the previous section. ion, we reduced its mass by a factor of sin(i) because in the context of second-order theory, it is appropriate to only consider the projection of its mass onto the disk’s reference plane. 4. Production of Highly Eccentric Planets Rewriting the above Hamiltonian in terms of cartesian coordi- 0 0 0 0 nates (q = i cos Ω , p = i sin Ω ), the first-order perturbation In the two preceeding sections, we have shown that in a binary equations (p ˙ = ∂K/∂q, q˙ = −∂K/∂p) yield an eigen-system stellar system, a self-gravitating disk avoids dynamical excita- that can be solved analytically (see Ch. 7 of Murray and Dermott tion, arising from the stellar companion, even if inclined. As (1999)). In our calculations, we choose N = 101 and again con- a result, one can expect that formation of planetary systems is sider a σ ∝ r−1 surface density across the disk. generally not inhibited. Furthermore, even after the disappear- We take the orbital properties of the stellar companion to be ance of the gas, we expect that the Kozai effect can continue to the same as those discussed in the previous section and take the be wiped out as a result of apsidal precession, induced by planet- Batygin et al.: Planet Formation in Inclined Binaries 7

Fig. 6. Phase-space plot of the inner planet, corresponding to Fig. 7. Domain of applicability of the arguments presented in the orbital evolution, shown in Figure 4. Prior to the instability this paper. The red curve shows the dividing line between disk- (t < 12Myr), the motion of the planet (shown as gray points) is dominated and stellar companion-dominated apsidal precession non-resonant. However, after a the outer planet gets ejected, the (as in section 2). The three purple curves illustrate the disap- remaining planet enters the Kozai resonance (shown as a black pearance of the Kozai separatrix, for various choices of maxi- line). mal inclination (as in section 3). The black curve delineates the boundary between rigid precession of the disk’s mid-plane and a warped structure (as in section 4). Successful formation of plan- planet interactions. As already discussed in the introduction, this ets can take place in well-separated binary systems where disk is the case for the planets of the solar system with respect to the self-gravity dominates over perturbations from the stellar com- galactic tide, or the satellites of Uranus relative to solar pertur- panion. bations. Consequently, the final issue we need of address is how planets, such as HD80606b and 16Cygni Bb, do eventually end up undergoing Kozai cycles. The evolutionary path that a planetary system can take between the birth nebula stage and the Kozai stage is necessarily circular orbits (e1 = e2 = 0.01) with a1 = 5AU and a2 = 7.5AU non-unique. One obvious possibility is that only a single large in the same plane. The stellar companion was taken to be on a ◦ planet forms in the disk and as the gas evaporates, self-gravity circulara ˜ = 1000AU orbit, inclined by i = 80 with respect of the disk becomes insufficient to wipe out the Kozai resonance. to the orbital plane of the planets. We performed the simulation Such a scenario, although possible, is probably far from being using a modified version of SyMBA (Duncan et al., 1998) in universal, since the observed multiplicity in planetary systems which a companion star is set on a distant, fixed circular orbit. (Mugrauer et al., 2010), as well as theoretical considerations The timestep was chosen to be 0.2y and throughout the integra- − (Armitage, 2010), suggest that protoplanetary disks rarely pro- tion, the fractional energy error remained below ∆E/E 6 10 5. duce only a single body. The system was evolved over 108 years. However, an alternative picture can be envisioned: a multi- The orbital evolution of the system is shown in Figure 5. As ple system forms and after the dispersion of the disk, still pro- can be seen, the system appears stable for the first ∼ 6 Myr, with tects itself from the Kozai cycles, exerted by the companion star, no sign of Kozai oscillations. However, at ∼ 6 Myr, the system through self-induced apsidal precession. Then, following a dor- becomes unstable because the planets‘ orbital proximity (initial mant period ,a dynamical instability occurs, removing all planets orbital separation is only ∼ 7 Hill radii) prevents them from re- except one, and therefore the remaining object starts to experi- maining stable on long timescales (Chambers et al., 1996). The ence the Kozai resonance. Such a scenario would be consider- eccentricities of the planetary orbits grow and eventually, the ably more likely, since dynamical instabilities are probably com- planets begin to experience close encounters with each other. mon among newly-formed planetary systems (Ford and Rasio, At ∼ 12 Myr, one of the two planets is ejected onto a hyper- 2008; Raymond et al., 2009). In particular, over the last decade bolic orbit. The remaining planet, now alone and with no apsidal or so, it has been realized that a transient dynamical instabil- precession, gets captured into the Kozai resonance, and starts ity has played an important role in shaping the architecture of undergoing large, coupled oscillations in eccentricity and incli- the solar system (Thommes et al., 1999; Tsiganis et al., 2005; nation. It is noteworthy that had the remaining planet ended up Morbidelli et al., 2007). Moreover, planet-planet scattering has on an orbit with smaller semi-major axis, general relativistic pre- been suggested to be an important process in explaining the ec- cession could have wiped out the Kozai effect (Wu and Murray, centricity distribution of extra-solar planets (Juric´ and Tremaine, 2003; Fabrycky and Tremaine, 2007). Additionally, although in 2008) as well as the misalignment of planetary orbits with stellar our setup, the stellar companion was initialized with a high in- rotation axes (Morton and Johnson, 2011). clination, this is not a necessary condition for the scenario, since The usefulness of analytical methods is limited when it the scattering event can generate mutual inclination. The tran- comes to the particular study of dynamical instabilities, so one sition from non-resonant motion to that characterized by Kozai must resort to numerical methods. Below, we demonstrate a nu- cycles is depicted in Figure 6, where orbital parameters prior to merical proof-of-concept of the scenario outlined above. the second planet’s ejection are shown as gray dots and the res- The system we considered was a pair of giant planets, both onant motion is shown as a black curve. Note the similarity of with mp = 1MJ around a sun-like (M? = 1M ) star, perturbed resonant motion computed numerically, to that computed analyt- by am ˜ = 2M companion. The planets were initialized on near ically, shown in Figure 3. 8 Batygin et al.: Planet Formation in Inclined Binaries

5. Discussion and Johnson, 2011). However, the model presented here suggests that the two distributions should be intimately related, as planet- In this paper, we have addressed the issue of how planetes- planet scattering provides the initial condition from which Kozai imals could preserve relative velocities that are slow enough cycles originate. Consequently, a quantitative re-examination of to allow planet accretion to take place, in binary stellar sys- the orbit-spin axis misalignment angle distribution, formed by tems. Particularly, we focused on highly inclined systems where Kozai cycles with tidal friction that originate from a scattered or- Kozai resonance with the perturbing stellar companion have bital architecture, and subsequent comparison of the results with been thought to disrupt the protoplanetary disk and inhibit planet observations of the Rossiter-McLaughlin effect will likely yield formation (Marzari et al., 2009). Here, we have shown, from new insights into dynamical evolution histories of misaligned analytical considerations, that fast apsidal precession, which re- hot Jupiters. sults from the disk self-gravity, wipes out the Kozai resonance Acknowledgments and ensures rigid precession of the disk’s nodal reference plane. We thank the referee, Y. Wu for useful suggestions. It is useful to consider the domain of applicability of the cri- teria discussed here. Namely, the trade-off between stellar binary separation and the perturbing companion’s mass should be quantified. The region of parameter space (binary separationa ˜ vs References disk mass to perturber mass ratio) where self-gravity suppresses secular excitation from the binary companion is delineated in Armitage, P. J. 2010. Astrophysics of Planet Formation. Astrophysics of Planet Formation, by Philip J. Armitage, pp. 294. ISBN 978-0-521-88745-8 (hard- Figure 7. The red curve shows the dividing line between disk- back). Cambridge, UK: Cambridge University Press, 2010. . dominated and stellar companion-dominated apsidal precession Bailey, M. E., Chambers, J. E., Hahn, G. 1992. Origin of sungrazers - A frequent (as in section 2). The three purple curves illustrate the disap- cometary end-state. Astronomy and Astrophysics 257, 315-322. pearance of the Kozai separatrix, for various choices of maxi- Binney, J., Tremaine, S. 1987. Galactic dynamics. Princeton, NJ, Princeton mal inclination (as in section 3). The black curve delineates the University Press, 1987, 747 p. . Chambers, J. E., Wetherill, G. W., Boss, A. P. 1996. The Stability of Multi-Planet boundary between rigid precession of the disk’s mid-plane and Systems. Icarus 119, 261-268. a warped structure (as in section 4). Correia, A. C. M., and 11 colleagues 2008. The ELODIE survey for northern As can be deduced from Figure 7, for distant stellar com- extra-solar planets. IV. HD 196885, a close binary star with a 3.7-year planet. panions (ã ∼ 1000 AU), the required total disk mass is of order Astronomy and Astrophysics 479, 271-275. Cochran, W. D., Hatzes, A. P., Butler, R. P., Marcy, G. W. 1997. The Discovery Mdisk ∼ 1-10MJ (depending on the perturber’s mass), consider- of a Planetary Companion to 16 Cygni B. The Astrophysical Journal 483, ably less than or comparable to, the total mass of the minimum 457. mass solar nebula. This implies that generally, protoplanetary Duquennoy, A., Mayor, M. 1991. Multiplicity among solar-type stars in the so- disks in binary stars can maintain roughly circular, unwarped lar neighbourhood. II - Distribution of the orbital elements in an unbiased sample. Astronomy and Astrophysics 248, 485-524. and untwisted structures. Consequently, we can conclude that Duncan, M. J., Levison, H. F., Lee, M. H. 1998. A Multiple Time Step planetary formation in wide binary systems is qualitatively no Symplectic Algorithm for Integrating Close Encounters. The Astronomical different from planetary formation around single stars. Journal 116, 2067-2077. After the formation of planets is complete and the gaseous Eggenberger, A., Udry, S., Mayor, M. 2003. Planets in Binaries. Scientific Frontiers in Research on Extrasolar Planets 294, 43-46. nebula has dissipated, the Kozai effect can continue to be inhib- Eggleton, P. P., Kiseleva-Eggleton, L. 2001. Orbital Evolution in Binary and ited as a result of orbital precession induced by planet-planet in- Triple Stars, with an Application to SS Lacertae. The Astrophysical Journal teractions. However, as the numerical experiment presented here 562, 1012-1030. suggests, if a planetary system experiences a transient dynamical Fabrycky, D., Tremaine, S. 2007. Shrinking Binary and Planetary Orbits by instability that leaves the planets on sufficiently well-separated Kozai Cycles with Tidal Friction. The Astrophysical Journal 669, 1298-1315. Ford, E. B., Rasio, F. A. 2008. Origins of Eccentric Extrasolar Planets: Testing orbits, the planets can start undergoing Kozai cycles. An evolu- the Planet-Planet Scattering Model. The Astrophysical Journal 686, 621-636. tionary sequence of this kind can explain the existence of orbital Fragner, M. M., Nelson, R. P. 2010. Evolution of warped and twisted accretion architectures characterized by highly eccentric planets, such as discs in close binary systems. Astronomy and Astrophysics 511, A77. those of HD 80606 and 16 Cygni B (Eggleton and Kiseleva- Fragner, M. M., Nelson, R. P., Kley, W. 2011. On the dynamics and collisional growth of planetesimals in misaligned binary systems. Astronomy and Eggleton, 2001; Wu and Murray, 2003). Astrophysics 528, A40. 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